Average Error: 0.2 → 0.2
Time: 32.7s
Precision: 64
\[\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{rand \cdot 1}{\sqrt{\frac{\left(a - \frac{1.0}{3.0}\right) \cdot \left(9 \cdot \left(a + \frac{1.0}{3.0}\right)\right)}{a + \frac{1.0}{3.0}}}}\right)\]
\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)
↓
\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{rand \cdot 1}{\sqrt{\frac{\left(a - \frac{1.0}{3.0}\right) \cdot \left(9 \cdot \left(a + \frac{1.0}{3.0}\right)\right)}{a + \frac{1.0}{3.0}}}}\right)
double f(double a, double rand) {
        double r2397087 = a;
        double r2397088 = 1.0;
        double r2397089 = /* ERROR: no posit support in C */;
        double r2397090 = 3.0;
        double r2397091 = /* ERROR: no posit support in C */;
        double r2397092 = r2397089 / r2397091;
        double r2397093 = r2397087 - r2397092;
        double r2397094 = 1.0;
        double r2397095 = /* ERROR: no posit support in C */;
        double r2397096 = 9.0;
        double r2397097 = /* ERROR: no posit support in C */;
        double r2397098 = r2397097 * r2397093;
        double r2397099 = sqrt(r2397098);
        double r2397100 = r2397095 / r2397099;
        double r2397101 = rand;
        double r2397102 = r2397100 * r2397101;
        double r2397103 = r2397095 + r2397102;
        double r2397104 = r2397093 * r2397103;
        return r2397104;
}
↓
double f(double a, double rand) {
        double r2397105 = a;
        double r2397106 = 1.0;
        double r2397107 = 3.0;
        double r2397108 = r2397106 / r2397107;
        double r2397109 = r2397105 - r2397108;
        double r2397110 = 1.0;
        double r2397111 = rand;
        double r2397112 = r2397111 * r2397110;
        double r2397113 = 9.0;
        double r2397114 = r2397105 + r2397108;
        double r2397115 = r2397113 * r2397114;
        double r2397116 = r2397109 * r2397115;
        double r2397117 = r2397116 / r2397114;
        double r2397118 = sqrt(r2397117);
        double r2397119 = r2397112 / r2397118;
        double r2397120 = r2397110 + r2397119;
        double r2397121 = r2397109 * r2397120;
        return r2397121;
}

Error

Bits error versus a

Bits error versus rand

Derivation

  1. Initial program 0.2

    \[\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)\]
  2. Using strategy rm

  3. Applied *p16-rgt-identity-expand0.2

    \[\leadsto \color{blue}{\left(\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(1.0\right)\right)} \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)\]

  4. Applied associate-*l*0.2

    \[\leadsto \color{blue}{\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\left(1.0\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)\right)}\]

  5. Simplified0.2

    \[\leadsto \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \color{blue}{\left(\frac{\left(1\right)}{\left(\frac{\left(rand \cdot \left(1\right)\right)}{\left(\sqrt{\left(\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(9\right)\right)}\right)}\right)}\right)}\]
  6. Using strategy rm

  7. Applied p16-flip--0.2

    \[\leadsto \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\frac{\left(rand \cdot \left(1\right)\right)}{\left(\sqrt{\left(\color{blue}{\left(\frac{\left(\left(a \cdot a\right) - \left(\left(\frac{\left(1.0\right)}{\left(3.0\right)}\right) \cdot \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}{\left(\frac{a}{\left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)}\right)}\right)} \cdot \left(9\right)\right)}\right)}\right)}\right)\]

  8. Applied associate-*l/0.2

    \[\leadsto \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\frac{\left(rand \cdot \left(1\right)\right)}{\left(\sqrt{\color{blue}{\left(\frac{\left(\left(\left(a \cdot a\right) - \left(\left(\frac{\left(1.0\right)}{\left(3.0\right)}\right) \cdot \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right) \cdot \left(9\right)\right)}{\left(\frac{a}{\left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)}\right)}\right)}}\right)}\right)}\right)\]

  9. Simplified0.2

    \[\leadsto \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\frac{\left(rand \cdot \left(1\right)\right)}{\left(\sqrt{\left(\frac{\color{blue}{\left(\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\left(9\right) \cdot \left(\frac{a}{\left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)}\right)\right)\right)}}{\left(\frac{a}{\left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)}\right)}\right)}\right)}\right)}\right)\]

  10. Final simplification0.2

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{rand \cdot 1}{\sqrt{\frac{\left(a - \frac{1.0}{3.0}\right) \cdot \left(9 \cdot \left(a + \frac{1.0}{3.0}\right)\right)}{a + \frac{1.0}{3.0}}}}\right)\]

Reproduce

herbie shell --seed 2019162 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (*.p16 (-.p16 a (/.p16 (real->posit16 1.0) (real->posit16 3.0))) (+.p16 (real->posit16 1) (*.p16 (/.p16 (real->posit16 1) (sqrt.p16 (*.p16 (real->posit16 9) (-.p16 a (/.p16 (real->posit16 1.0) (real->posit16 3.0)))))) rand))))